The solvability of groups with nilpotent minimal coverings

TL;DR

This paper proves that any group with a nilpotent minimal covering must be solvable, extending the understanding of group coverings and their implications for group structure.

Contribution

It completes the proof that groups with nilpotent minimal coverings are solvable, building on prior results about minimal counterexamples being almost simple finite groups.

Findings

Groups with nilpotent minimal coverings are solvable.

Minimal counterexamples are almost simple finite groups.

The proof completes the classification of such groups.

Abstract

A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a proof that every group that has a nilpotent minimal covering is solvable, starting from the previously known result that a minimal counterexample is an almost simple finite group.